Radiation condition enforcement in linear free-surface wave modeling

Linear free-surface wave models must satisfy the radiation condition to avoid unphysical solutions. The radiation condition requires that the wave amplitude vanishes at infinite boundaries and waves propagate outward from the source location. The former requirement has received much attention in the literature, but the latter is often ignored. Linear-free surface waves at the deep-water condition are defined by dispersion curves that include two V shaped curves and an additional ring curve when the pulsation frequency is below a certain value. Constraints that enforce both requirements of the radiation condition have been developed and applied to wave solutions associated with each dispersion curve. The two requirements of the radiation condition produce the same constraint for the ring curve and one V shaped curve referred to as the Inner V curve. However, for the second V shaped curve referred to as the Outer V curve, satisfying the requirement of vanishing wave amplitude at infinite boundaries does not guarantee the outgoing condition. When the ring curve exists, this Outer V curve only generates ingoing waves. At higher pulsation frequencies, the dimensionless wavenumber component in the source moving direction should be smaller than the dimensionless frequency to satisfy both requirements of the rdiation conditions. Thus, models that only satisfy vanishing amplitude at infinite boundaries would include ingoing waves associated with this Outer V curve. The wave patterns that include ingoing waves are found to be oversensitive to a small deviation from the steady source motion. The ingoing waves may also affect predictions of the wave amplitude and the wave resistance of the ship.